Probability that $~r_{0}(n)>n/k$

Question

For a positive integer greater than $1$, let, under Goldbach's conjecture, $r_{0}(n):=\inf\{r>0, (n-r,n+r)\in\mathbb{P}^{2}\}$. What is the probability $P_{k}(n)$ that $r_{0}(n)>n/k$ where $k$ is a positive real number greater than $1$? Does $\lim_{x\to\infty}\sum_{n\leq x}P_{k}(n)$ exist for all such $k$? Is it always less than $1$?

Answer 1

Given that:

• if $n$ is 3 mod 6, $r$ needs to be 2 or 4 mod 6
• if $n$ is 0 mod 6, $r$ needs to be 1 or 5 mod 6
• if $n$ is 2 or 4 mod 6, $r$ needs to be 3 mod 6
• if $n$ is 1 or 5 mod 6, $r$ needs to be 0 mod 6

or $2n-3$ is prime. $k$ would get lower as we ruled small cases out.

As to the limit I have no clue, except $k$ needs be less than $n\over 2$ unless $n$ is 0 mod 6.